Post on 02-Feb-2021
Image Reconstruction 3 – Fully 3D Reconstruction
Thomas Bortfeld
Massachusetts General Hospital, Radiation Oncology, HMS
HST.S14, February 25, 2013
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 1 / 26
Outline
1 The 3D Radon transform and its inverseThe 3D Radon transformInversion of the 3D Radon transform
2 Getting 3D Radon transform from cone beam data (1990s)Grangeat’s trick
3 Tuy theorem
4 Helical scanning
5 The Katsevich breakthrough (2002)
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 2 / 26
The 3D Radon transform and its inverse The 3D Radon transform
3D Radon transform
The 3D Radon transform of f(x) is theintegral of f(x) over 2D planesperpendicular to n̂Ω
Rf(p, n̂Ω) =
∫V
f(x) δ(p− x · n̂Ω) d3x
1 Note: The δ-function “picks” thosepoints x that lie on the plane shown(plane at distance p from origin).
2 Note: In 3D (and higher dimensions)the Radon transform differs from the”x-ray transform” (integration overlines).
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 3 / 26
The 3D Radon transform and its inverse Inversion of the 3D Radon transform
The inverse 3D Radon transform
Inversion of the Radon transform in 3D is beautifully simple:
f(x) = − 18π2
∫4π
R′′f(x · n̂Ω, n̂Ω) dΩ,
where
R′′f(p, n̂Ω) =∂2R′′f(p, n̂Ω)
∂p2.
Steps involved:
1 take second derivative (with respect to p) of Radon transform,
2 back-project over planes containing x,
3 integrate over all angles.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 4 / 26
The 3D Radon transform and its inverse Inversion of the 3D Radon transform
The inverse 3D Radon transform
Inversion of the Radon transform in 3D is beautifully simple:
f(x) = − 18π2
∫4π
R′′f(x · n̂Ω, n̂Ω) dΩ,
where
R′′f(p, n̂Ω) =∂2R′′f(p, n̂Ω)
∂p2.
Steps involved:
1 take second derivative (with respect to p) of Radon transform,
2 back-project over planes containing x,
3 integrate over all angles.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 4 / 26
The 3D Radon transform and its inverse Inversion of the 3D Radon transform
Intuitive ”derivation” of inverse 3D Radon transform
Central slice theorem in 3D: every back-projection (“smearing out”over planes) for a given orientation n̂Ω corresponds with adding a linethrough the origin of the Fourier domain.
The density of lines falls off as 1/ρ2.
To compensate for the low-pass effect, we must multiply with ρ2
(compare with the |ρ| = |ν| filter in the 2D case).Multiplication with ρ2 in the Fourier domain corresponds with takingthe second derivative in the spatial domain.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 5 / 26
The 3D Radon transform and its inverse Inversion of the 3D Radon transform
Intuitive ”derivation” of inverse 3D Radon transform
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 6 / 26
The 3D Radon transform and its inverse Inversion of the 3D Radon transform
So what is the problem?
The problem is that with x-rays we can only measure line integrals(along the rays), not the planar integrals needed for the 3D Radontransform.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 7 / 26
Getting 3D Radon transform from cone beam data (1990s)
Outline
1 The 3D Radon transform and its inverseThe 3D Radon transformInversion of the 3D Radon transform
2 Getting 3D Radon transform from cone beam data (1990s)Grangeat’s trick
3 Tuy theorem
4 Helical scanning
5 The Katsevich breakthrough (2002)
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 8 / 26
Getting 3D Radon transform from cone beam data (1990s)
Getting 3D Radon transform from cone beam data
The problem is that with x-rays we can only measure line integrals(along the rays), not the planar integrals needed for the 3D Radontransform.
But, can’t we just integrate line integrals across the plane to get theplanar integral?
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 9 / 26
Getting 3D Radon transform from cone beam data (1990s)
Getting 3D Radon transform from cone beam data
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 10 / 26
Getting 3D Radon transform from cone beam data (1990s)
Getting 3D Radon transform from cone beam data
Measured line integrals:
∞∫0
f (x(L, γ, α)) dL
Integrate over the plane:
π∫−π
∞∫0
f (x(L, γ, α)) dLdγ =
π∫−π
∞∫0
1
Lf (x(L, γ, α))LdLdγ
6= Rf(p, n̂Ω)
This would give the integral of 1Lf over the plane.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 11 / 26
Getting 3D Radon transform from cone beam data (1990s)
Grangeat’s trick
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 12 / 26
Getting 3D Radon transform from cone beam data (1990s) Grangeat’s trick
Grangeat’s ”trick”
Grangeat’s idea: Compensate for incorrect integration over the planeby taking an incorrect derivative of the Radon transform with respectto p.
Apply a weight factor of 1/ cos γ and take the derivative with respect tothe tilt angle, α:
π∫−π
∂
∂α
1
cos γ
∞∫0
f (x(L, γ, α)) dLdγ =
π∫−π
∂
∂p
L cos γ
cos γ
∞∫0
f (x(L, γ, α)) dLdγ
=∂
∂p
π∫−π
∞∫0
f (x(L, γ, p))LdLdγ
=∂Rf(p, n̂Ω)
∂p.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 13 / 26
Getting 3D Radon transform from cone beam data (1990s) Grangeat’s trick
Problems with Grangeat-type methods
“Long object” problem – solvable, approximately
Numerical instabilities
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 14 / 26
Tuy theorem
Tuy’s sufficiency condition
Theorem (Tuy)
Any plane through an object point x must cut the source trajectory.a
aH.K.Tuy, An inversion formula for cone-beam reconstruction, SIAM J Appl.Math., 43(3), 546-552, 1983. See also A.A. Kirillow, On a problem of I. M.Gel’fand, Soviet Math., 2 (1961), pp. 268-269.
Condition for trajectory to facilitate exact reconstruction
Examples of source trajectories – which ones fulfill the Tuy condition?
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 15 / 26
Helical scanning
Helical scanning geometry
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 16 / 26
Helical scanning
Helical scanning geometry: Resort into planes
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 17 / 26
The Katsevich breakthrough (2002)
Outline
1 The 3D Radon transform and its inverseThe 3D Radon transformInversion of the 3D Radon transform
2 Getting 3D Radon transform from cone beam data (1990s)Grangeat’s trick
3 Tuy theorem
4 Helical scanning
5 The Katsevich breakthrough (2002)
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 18 / 26
The Katsevich breakthrough (2002)
The Katsevich breakthrough
Filtered backprojection algorithm for cone beam data
Exact reconstruction (aside from discretization)
Good for “long objects” (projection data only needed near the ROI)
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 19 / 26
The Katsevich breakthrough (2002)
The Katsevich algorithm
Detector signal and its derivative:
D(y(s),θ)
D′(y(s),θ) =∂D(y(q),θ)
∂q
∣∣∣∣q=s
Further definitions:
b =x− y(s)|x− y(s)|
; y(s): source position
e: perpendicular to b, must be appropriately chosen
b and e span the κ plane for filtering
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 20 / 26
The Katsevich breakthrough (2002)
The Kappa plane
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 21 / 26
The Katsevich breakthrough (2002)
The Kappa plane
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 22 / 26
The Katsevich breakthrough (2002)
The Katsevich algorithm
f(x) = − 12π2
∫IPI(x)
I(s,x)
|x− y(s)|ds
where
I(s,x) =
π∫−π
1
sin γD′(y(s), cosγ b+ sinγ e) dγ
Note:
1/ sin γ is the filter, it corresponds to a “Hilbert” filter.
Backprojection is done over the PI (parametric interval), seeDanielsson et al.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 23 / 26
The Katsevich breakthrough (2002)
Results: 3D Shepp phantom – from Katsevich 2002
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 24 / 26
The Katsevich breakthrough (2002)
Results: Defrise slice phantom – from Katsevich 2002
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 25 / 26
Further reading on fully 3D reconstruction
A. Katsevich: Theoretically exact filtered backprojection-typeinversion algorithm for spiral CT. SIAM J. Appl. Math.62(6):2012-2026, 2002.
A. Katsevich: Analysis of an exact inversion algorithm for spiralcone-beam CT. Phys. Med. Biol. 47:2583-2597, 2002.
C. Bontus, T. Köhler: Reconstruction Algorithms for ComputedTomography. Advances in Imaging and Electron Physics 151:1-63,2008.
Papers by P.-E. Danielsson et al., Linköping, Sweden
F. Natterer: The Mathematics of Computerized Tomography.Reprint: SIAM Classics in Applied Mathematics, 2001.
H.K. Tuy: An inversion formula for cone-beam reconstruction. SIAMJ Appl. Math., 43(3), 546-552, 1983.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 3 – Fully 3D ReconstructionHST.S14, February 25, 2013 26 / 26
1The 3D Radon transform and its inverseThe 3D Radon transformInversion of the 3D Radon transform
Getting 3D Radon transform from cone beam data (1990s)Grangeat's trick
Tuy theoremHelical scanningThe Katsevich breakthrough (2002)
2